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army1987 comments on Open Thread, April 15-30, 2013 - Less Wrong
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BTW, how comes the ring with one element isn't usually considered a field?
The theorems work out nicer if you don't. A field should be a ring with exactly two ideals (the zero ideal and the unit deal), and the zero ring has one ideal.
Ah, so it's for exactly the same reason that 1 isn't prime.
Yes, more or less. On nLab this phenomenon is called too simple to be simple.
We often want the field without zero to form a multiplicative group, and this isn't the case in the ring with one element (because the empty set lacks an identity and hence isn't a group). Indeed we could take the definition of a field to be
and this is fairly elegant.